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Answer:

To find the value of [1 / alpha^3 + 1 / beta^3], we can use the relationship between the sum and product of roots of a quadratic polynomial.

Let's denote the sum of the roots as S and the product of the roots as P.

For the quadratic polynomial ax^2 + bx + c, the sum of the roots (S) is given by: S = -b / a

And the product of the roots (P) is given by: P = c / a

In your case, the polynomial is 4x^2 - 3x + 5, so a = 4, b = -3, and c = 5.

The sum of the roots (alpha and beta) is: S = -(-3) / 4 = 3/4

The product of the roots (alpha and beta) is: P = 5 / 4

Now, we can use the following identities:

1. alpha^3 + beta^3 = (alpha + beta)^3 - 3(alpha + beta)(alpha beta)

2. (alpha + beta) = S = 3/4

3. (alpha beta) = P = 5/4

Substitute these values into the first identity:

alpha^3 + beta^3 = (3/4)^3 - 3(3/4)(5/4)

alpha^3 + beta^3 = 27/64 - 45/64

alpha^3 + beta^3 = -18/64

alpha^3 + beta^3 = -9/32

Finally, the value of [1 / alpha^3 + 1 / beta^3] is 1 / (-9/32), which simplifies to -32/9.